Overview
This study focuses on properties of solutions to the Cauchy problem for time-fractional nonlinear diffusion equations. Specifically, it addresses the existence and uniqueness of $L^1$-solutions for time-fractional porous medium type equations. Additionally, the research investigates mass conservation for $L^1$-solutions in time-fractional fast diffusion equations and examines the phenomenon of finite-time extinction.
Research Context
The investigation centers on time-fractional nonlinear diffusion equations. These are a class of partial differential equations that incorporate fractional derivatives with respect to time, often used to model anomalous diffusion processes where the mean squared displacement of particles scales non-linearly with time. The specific types of equations considered are referred to as 'porous medium type' and 'fast diffusion' equations, which are well-known in classical diffusion theory and describe different regimes of diffusion behavior based on the nonlinearity.
Approach
The research establishes the global existence and uniqueness of $L^1$-solutions. This implies demonstrating that a solution exists for all times ($t>0$) and that this solution is the only one satisfying the given initial conditions within the $L^1$ function space. For time-fractional fast diffusion equations, the approach includes formulating and verifying a mass conservation law. Furthermore, the methodology extends to analyzing the long-term behavior of these solutions, specifically addressing whether finite-time extinction occurs for non-negative $L^1$-solutions.
Findings
- Global existence and uniqueness of $L^1$-solutions are established for the Cauchy problem associated with time-fractional porous medium type nonlinear diffusion equations.
- The mass conservation law is given for $L^1$-solutions to time-fractional fast diffusion equations.
- It is proven that finite-time extinction does not occur for any non-negative $L^1$-solutions of time-fractional fast diffusion equations.